Landau quantization of a circular Quantum Dot using the BenDaniel-Duke boundary condition
HHighlights
Landau quantization of a circular Quantum Dot using the BenDaniel-Duke boundary condi-tion
Sriram Gopalakrishnan,Sayak Biswas,Shivam Handa• The energy levels of a circular Quantum Dot (QD) under a transverse magnetic field, incorporating the Ben-Daniel Duke boundary condition (BDD) are derived and calculated numerically.• Theoretical findings were compared with the previously published experimental results on the GaAs-InGaAsQuantum Dot and found to be in agreement.• An insightful asymptotic approximation is provided, which converges with numerical results for larger valuesof size and confinement. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p andau quantization of a circular Quantum Dot using the BenDaniel-Duke boundary condition Sriram Gopalakrishnan, Sayak Biswas and Shivam Handa
Indian Institute of Technology Madras, Chennai, India- 600036Indian Institute of Science Education and Research, Kolkata, India - 741246Massachusetts Institute of Technology, Cambridge MA 02139
A R T I C L E I N F O
Keywords :HeterostructuresQuantum DotsEffective Mass TheoryBen-Daniel Duke boundary condition
A B S T R A C T
We derive the energy levels of a circular Quantum Dot (QD) under a transverse magnetic field,incorporating the Ben-Daniel Duke boundary condition (BDD). The parameters in our model arethe confinement barrier height, the size of the QD, the magnetic field strength, and a mass ratiohighlighting the effect of using BDD. Charge densities, transition energies, and the dependenceof energies on magnetic field has been calculated to show the strong influence of BDD. Wefind that our numerical calculations agree well with experimental results on the GaAs-InGaAsQuantum Dot and can be used further. We also provide an insightful analytical approximationto our numerical results, which converges well for larger values of size and confinement.
1. INTRODUCTION
Low dimensional quantum systems constitute an active area of research with widespread applications in technology.The non-abelian anyon, for instance, is a two dimensional (2D) quasiparticle proposed as an option for fault-tolerantquantum computation [1, 2]. Quantum Dots (QDs), also known as artificial atoms [3], are nanostructures that tightlyconfine electrons in quantum wells, resulting in bound states. Owing to their tunability, QDs have several applications,including quantum information processing [4, 5], and QD based light emitting devices [6, 7]. A recent experiment alsoprobed the energy levels of a QD in Bilayer Graphene [8]. Accurate level schemes of QDs are relevant in the presentcontext as technology inches towards quantum computing.Quantum Dots are fabricated by forming heterojunctions between dissimilar semiconductors [9]. It may be notedthat the core and shell materials of a heterojunction QD can have very similar properties if not for their bandgaps.Depending on the alignment of the valence and conduction band edges across the interface, QDs are classified asType-I or Type-II [10]. The nature of band edge alignment results in a confinement potential whose profile is usuallyapproximated as a parabola or a finite hard-wall for simplicity of theoretical modeling. Although a parabolic profileis less idealized than a finite hard-wall, the latter allows us to develop a phenomenology accounting for finite size andbarrier height.Additionally, models must employ Effective Mass Theory (EMT) accurately, so as to account for a spatially varyingcarrier effective mass created by the confinement potential. Hamiltonians must be modified to maintain hermiticity. Inthe case of hard-wall confinement, there is a discontinuous change in effective mass across the barrier. The correctedHamiltonian thus leads to a modified boundary condition on the derivative of the wavefunction, called the BenDanielDuke boundary condition (BDD) [12]. In this regard, we define a dimensionless mass ratio 𝛽 = 𝑚 𝑖 ∕ 𝑚 𝑜 , where 𝑚 𝑖 and 𝑚 𝑜 are the effective masses of the electron inside and outside the well respectively.The goal of this paper is to analytically develop the complete set of spin-degenerate Landau levels of a single-electron, hard-wall confined, circular QD placed in a perpendicular magnetic field using BDD and examining theeffect of imposing BDD. There have been extensive studies on QDs in the past three decades. However, only few ofthese models include and examine the effect of imposing BDD [13, 14, 15, 16, 17]. A recent theoretical work modelledCdSe/CdS core-shell QDs using BDD [18]. The general approach of our theory can be used to understand data obtainedin experiments such as Gated Transport Spectroscopy (GTS) and Single Electron Capacitance Spectroscopy (SECS)of Quantum Dots with electrostatic confinement and magnetic fields [3].The system we consider is a single electron trapped in a finite, radially symmetric potential well in 2D, and placedin a perpendicular magnetic field. We have accomodated the possibility of different magnetic fields inside and outsidethe QD, although we use a uniform magnetic field in numerical calculations. The confinement potential approximatesa thin InGaAs quantum disk sandwiched between two layers of GaAs, as experimentally probed by Drexler et. al. ORCID (s):
First Author et al.:
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Page 1 of 9 hort Title of the Article [21]. A hard-wall confinement model was soon proposed by Peeters et. al. [22], however, they used a two-electronmodel without BDD to fit a transition gap with experimental data. It is known in practice that the transition gapsof a QD are effectively independent of electron-electron interactions [23, 24]. Hence, we propose a single-electronmodel in conjunction with BDD to find agreement with the same data. We would like to mention that the ground stateof our system was studied, including the effect of BDD, by Asnani et. al. [25]. We extend the study using Landauquantization to obtain a complete electronic structure from the Schrodinger equation, and test the effect of imposingBDD on multiple Landau levels created by a homogeneous magnetic field.QDs are also of interest from a fundamental physics point of view, particularly in understanding non-local phe-nomena such as the Aharanov-Bohm effect, where charges can be influenced by electromagnetic potentials even in theabsence of electromagnetic fields [26, 27, 28]. In the following analysis, although we consider an inhomogeneous mag-netic field only to keep the formalism general, there is an interesting phenomenon of magnetic edge states, where thereis additional quantization in terms of "missing flux quanta" [29]. It would be interesting to explore these phenomenain the context of BDD in future work.The paper is organized as follows. In Sec. II, we present our mathematical model of the system in interest. Thisinvolves setting up the Hamiltonian, solving for the wavefunction, and applying boundary conditions. In Sec. III, wedevelop an asymptotic approximation to the energy levels of the QD. In Sec. IV, we discussion the results we obtained,including experimental agreement and the validity of our approximation, followed by concluding remarks.
2. Model
The QD is modeled as an electron trapped in a cylindrical potential well of radius R and barrier height 𝑉 in a 2Dplane. In cylindrical polar coordinates ( 𝑟, 𝜙, 𝑧 ) , the lateral confinement potential used is given by 𝑉 ( 𝑟 ) = { 𝑟 ≤ 𝑅𝑉 𝑜 𝑟 > 𝑅 (1)In a realistic setting, the QD is also confined along the z-axis due to a cylindrical or lens shape. However, the energyassociated with vertical confinement is much larger, and is decoupled from lateral confinement for transition gapmeasurements in the experiment we are interested in [21]. Note that Equation (1) represents an idealized hard-wallelectrostatic confinement, but is better than a parabolic profile as it accounts for the finite lateral size and barrier heightof the QD. The QD is placed in a perpendicular magnetic field, which takes a uniform value 𝐵 𝑖 inside the QD, and 𝐵 𝑜 outside the QD respectively. ⃗𝐵 ( 𝑟 ) = { 𝐵 𝑖 ̂𝑧 𝑟 ≤ 𝑅𝐵 𝑜 ̂𝑧 𝑟 > 𝑅 (2)Note that we consider an inhomogeneous magnetic field only to keep the analysis general for potential future work.Calculations based on the model presented in Section 4 assume a homogeneous magnetic field: 𝐵 𝑖 = 𝐵 𝑜 = 𝐵 . Themagnetic field profile corresponds to a continuous magnetic vector potential ⃗𝐴 ( 𝑟 ) given by ⃗𝐴 𝑝 ( 𝑟 ) = [ 𝐵 𝑝 𝑟 𝑝 𝑟 ] ̂𝜙, 𝑝 = 𝑖 or 𝑜 (3)The subscript ’ 𝑝 ’ can be either ’ 𝑖 ’(inside) or ’ 𝑜 ’(outside), and is helpful in generalizing the analysis inside and outsidethe QD. In Equation (3), Φ 𝑝 is an intermediate variable with the dimensions of magnetic flux, defined as Φ 𝑝 = { p = i ( 𝐵 𝑖 − 𝐵 𝑜 ) 𝑅 p = o (4)The Hamiltonian of the system is that of an electron placed in an electromagnetic field [30] ̂𝐻 = 12 𝑚 𝑝 [ ̂𝑝 + 𝑒 ⃗𝐴 ] + 𝑉 ( 𝑟 ) (5) First Author et al.:
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The Hamiltonian commutes with 𝜕 ∕ 𝜕𝜙 , and hence the wavefunction is separable as 𝜓 ( 𝑟, 𝜙 ) = 𝑒 𝑖𝑙𝜙 𝑔 ( 𝑟 ) where 𝑙 is aninteger (0, ±1, ±2 . . . ) The radial part 𝑔 ( 𝑟 ) of the time independent Schrodinger equation ̂𝐻𝜓 = 𝐸𝜓 is hence foundto be − 𝐾 𝑝,𝑙 = 1 𝑔 ( 𝑔 ′′ + 𝑔 ′ 𝑟 ) − 𝑙 𝑟 − 𝑒 ℏ | ⃗𝐴 ( 𝑟 ) | − 2 𝑙𝑒𝑟ℏ | ⃗𝐴 ( 𝑟 ) | (6)where we define a wave vector 𝐾 𝑝 as 𝐾 𝑝,𝑙 = ⎧⎪⎨⎪⎩ 𝑚 𝑖 𝐸ℏ inside 𝑚 𝑜 ℏ ( 𝐸 − 𝑉 𝑜 ) outside (7)The solution of Eq. (6) leads to an exact solution in terms of Kummer functions [31]. We find, however, that thesolution can be well approximated in terms of Bessel functions in a regime defined by constraints on the size andbarrier height of the QD, 𝑅 ≪ √ ℏ𝑒𝐵 = √ 𝐿 𝑚 (8) 𝑉 𝑜 ≫ ℏ 𝑚 𝑜 𝑅 (9)Here 𝐿 𝑚 = √ ℏ ∕ 𝑒𝐵 is the magnetic length scale and is also called the Landau length. For 𝐵 = 1 T, we require
𝑅 ≪ nm. This is reasonable since the radii involved in the experiments of Drexler et. al. was nm. Further, if 𝑅 = 10 nmwe require 𝑉 𝑜 ≫ . meV (using 𝑚 𝑜 = 0 . 𝑚 𝑒 ), which is a modest lower bound when we use barrier heights of theorder of several hundreds of meV or a few eV (around 100 meV in case of the InGaAs-GaAs QD).Under the regime defined by Eq. (8) and Eq. (9), we find that the radial wavefunction can be approximated as 𝑔 𝑙𝑖 ( 𝑟 ) = 𝐴 exp ( − 𝑒𝐵 𝑖 𝑟 ℏ ) 𝐽 𝑙 ( 𝑘 𝑖 𝑟 ) (10) 𝑔 𝑙𝑜 ( 𝑟 ) = 𝐵 exp ( − 𝑒𝐵 𝑜 𝑟 ℏ ) exp(− 𝑘 𝑜 𝑟 ) √ 𝑟 (11)Here 𝐽 𝑙 is the 𝑙 𝑡ℎ Bessel function of the first kind. We have also defined wave vectors 𝑘 𝑖 and 𝑘 𝑜 as 𝑘 𝑖,𝑙 = 2 𝑚 𝑖 𝐸ℏ − (2 𝑙 + 1) 𝑒𝐵 𝑖 ℏ (12) 𝑘 𝑜,𝑙 = 2 𝑚 𝑜 ℏ ( 𝑉 𝑜 − 𝐸 ) + (2 𝑙 + 1) 𝑒𝐵 𝑜 ℏ + 𝑒 𝑅 ℏ 𝐵 𝑜 ( 𝐵 𝑖 − 𝐵 𝑜 ) (13)Note the last term in Eq. (13). If 𝐵 𝑖 = 𝐵 𝑜 , the two expressions (Eqs. (12) and (13)) are the same given the shift inenergy 𝐸 to 𝑉 𝑜 − 𝐸 .We now apply boundary conditions to our solution (Eq. (10) and Eq.(11)) to obtain quantized energy levels.Although the wavefunction is continuous at 𝑟 = 𝑅 , its derivative is discontinuous. 𝑔 𝑙𝑖 ( 𝑅 ) = 𝑔 𝑙𝑜 ( 𝑅 ) (14) 𝑑𝑔 𝑙𝑖 𝑑𝑟 |||| 𝑟 = 𝑅 = 𝛽 𝑑𝑔 𝑙𝑜 𝑑𝑟 |||| 𝑟 = 𝑅 (15)Equation (15) is the Ben-Daniel Duke boundary condition for our system. As mentioned earlier, 𝛽 = 𝑚 𝑖 ∕ 𝑚 𝑜 is the ratioof effective masses inside and outside the well. Eliminating normalization constants, we obtain a non-linear equationfor the quantized energy levels of the system. 𝛽 𝛽𝑘 𝑜 𝑅 + 𝑒𝑅 ℏ ( 𝛽𝐵 𝑜 − 𝐵 𝑖 ) + 𝑘 𝑖 𝑅 𝐽 ′ 𝑙 ( 𝑘 𝑖 𝑅 ) 𝐽 𝑙 ( 𝑘 𝑖 𝑅 ) = 0 (16)Equation (16) cannot be solved analytically to obtain energy eigenvalues 𝐸 . However, one can obtain a simple asymp-totic approximation for the energy levels, which is the subject of our next section. First Author et al.:
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3. ASYMPTOTICS
Consider Eq. (16) for 𝑔 𝑖 ( 𝑟 ) , the radial wavefunction inside the QD. For a sufficiently large barrier height, we expect 𝑔 𝑖 ( 𝑟 ) to be close to zero at 𝑟 = 𝑅 . Equivalently, we expect the argument of 𝐽 𝑙 to be close to one of its nodes. 𝑘 𝑖 𝑅 = 𝑧 𝑛𝑙 − 𝜀 (17)Here 𝑧 𝑛𝑙 is the 𝑛 𝑡ℎ node of 𝐽 𝑙 and | 𝜀 | ≪ . We can then Taylor approximate 𝐽 𝑙 ( 𝑘 𝑖 𝑅 ) as 𝐽 𝑙 ( 𝑘 𝑖 𝑅 ) = 𝐽 𝑙 ( 𝑧 𝑛𝑙 − 𝜀 ) ≈ − 𝜀𝐽 ′ 𝑙 ( 𝑧 𝑛𝑙 ) (18)Hence we have 𝐽 ′ 𝑙 ( 𝑘 𝑖 𝑅 ) 𝐽 𝑙 ( 𝑘 𝑖 𝑅 ) ≈ − 1 𝜀 (19)Using this result in Eq. (16), we obtain an expression for 𝜀 , 𝜀 = 𝑧 𝑛𝑙 𝛽 + 𝛽𝑘 𝑜 𝑅 + 𝑒𝑅 ℏ ( 𝛽𝐵 𝑜 − 𝐵 𝑖 ) = 𝑧 𝑛𝑙 √ 𝜎 (20)For large 𝑉 𝑜 ( 𝛽𝑘 𝑜 𝑅 ≫ ), the largest term in the denominator of Eq. (20) is 𝛽𝑘 𝑜 𝑅 , where 𝑘 𝑜 ≈ √ 𝑚 𝑜 𝑉 𝑜 ℏ . Therefore 𝜎 ,as defined in Eq. (20) is approximately given by 𝜎 ≈ 2 𝑚 𝑜 ℏ ( 𝛽 𝑅 𝑉 𝑜 ) (21)Using Eq. (17) and Eq. (20) in conjunction with Eq. (12) for 𝑘 𝑖 , we have an asymptotic approximation for the energylevels as 𝐸 𝑛,𝑙 = ℏ 𝑧 𝑛𝑙 𝛽𝑚 𝑜 𝑅 ( √ 𝜎 ) + ( 𝑙 + 12 ) ℏ𝑒𝐵 𝑖 𝛽𝑚 𝑜 (22)Equation (22) has an elegant physical meaning. Suppose we had an electron trapped in cylindrical potential well withradius ( 𝑅 + 𝛿 ) , where 𝛿 = 𝑅 √ 𝜎 , without an external magnetic field. The first term of Eq. (22) represents quantizedenergy levels of the aforementioned system. If we now switch on a perpendicular magnetic field 𝐵 𝑖 inside the well,additional Landau levels are observed whose splitting energy is described by the second term of Eq. (22). 𝛿 can thusbe interpreted as a penetration depth of the wavefunction due to lateral confinement.The levels are hence classified by quantum numbers ( 𝑛, 𝑙 ) . The ground state of the QD is (1,0), while the next fivestates are (1,-1), (1,1), (1,-2), (1,2) and (2,0). The series is generated from the relative locations of 𝑧 𝑛𝑙 , the 𝑛 𝑡ℎ root of theBessel 𝐽 𝑙 ( 𝑥 ) , which displays the following trend: 𝑧 < 𝑧 < 𝑧 < 𝑧 . In the presence of a perpendicular magneticfield, the states ( 𝑛, 𝑙 ) and ( 𝑛, − 𝑙 ) lose their degeneracy, resulting in Landau level splitting with a gap Δ 𝐸 = 2 𝑙ℏ𝜔 𝑖 where 𝜔 𝑖 = 𝑒𝐵 𝑖 ∕ 𝑚 𝑖 . We also expect an additional but smaller, 𝑔𝜇 𝐵 𝐵 splitting between spin-up and spin-down electron states,well known as Zeeman splitting. We however ignore Zeeman splitting, and only consider the effect of BDD on spin-degenerate Landau levels.For 𝑉 = 100 meV, 𝑚 𝑜 = 0 . 𝑚 𝑒 , 𝛽 = 0 . , 𝑅 = 11 nm, homogeneous magnetic field 𝐵 𝑖 = 𝐵 𝑜 = 𝐵 , andconsidering the states 𝑛 = 1 , 𝑙 = ±1 , the asymptotic approximation reads 𝐸 ,𝑙 =±1 = [ . ( 𝐵 ) + ( 𝑙 + 12 ) . 𝐵 ] 𝑚𝑒𝑉 (23)where 𝐵 is in Tesla. Notice that the quadratic dependence on 𝐵 is negligibly small for the range of magnetic fieldwe are interested in and even beyond. This is why we do not see a curvature in the Landau levels (Figure 3) even forhigh magnetic fields. For 𝑅 = 10 nm, the Bessel approximation (Equation (10)) is applicable only when 𝐵 ≪ T. Weexpect the Bessel approximation to gradually break down for
𝐵 > T. For higher magnetic fields, the experimentallyexpected curvature in the Landau levels can only be observed by deriving the energy eigenvalues using the generalsolution to Equation (6) in terms of Kummer functions, which are notorious to deal with. For the purpose of this work,we limit ourselves to the effect of BDD on these approximately linear levels.
First Author et al.:
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4. RESULTS
The radial charge density of the QD is given by ̃𝜌 ( 𝑟 ) = 𝑒 𝜋𝑟𝑔 ( 𝑟 ) so that ∫ ̃𝜌 ( 𝑟 ) 𝑑𝑟 = 𝑒 . We investigated the scaledcharge density profile 𝜌 ( 𝑟 ) = 𝑟𝑔 ( 𝑟 ) of a QD with radius 10 nm, confined with 𝑉 𝑜 = 1 eV, and under a magnetic fieldof 1 T.In Fig. 1, we examine the effect of 𝛽 on the ground state charge density profile. We observe that charge spreads outcloser to the boundary as 𝛽 is decreased. Further, a small amount of charge leaks out of the QD for small 𝛽 and thisis enhanced as 𝛽 is decreased. This is an interesting observation, and can be explained by the fact that the tunnellingprobability at the boundary of the QD decays exponentially with the difference ( 𝑉 𝑜 − 𝐸 ) . As 𝛽 decreases, energy levelsrise and the tunnelling probability increases. One can also see the discontinuity in the derivative of 𝜌 ( 𝑟 ) at the boundaryfor small 𝛽 , which results from imposing BDD.The inset in Fig. 1 shows the charge density profile of states (1,0), (1,1) and (1,2). The states ( 𝑛, 𝑙 ) and ( 𝑛, − 𝑙 ) have the same charge density since they only differ by an overall phase ( 𝑒 𝑖 𝑙𝜙 ). We note that higher energy levels areassociated with more spreading of charge towards the boundary. The peak charge density is larger, and is closer to theboundary as we consider higher levels. Energy difference is the object of study in absorption and emission spectra. Hence, in Fig. (2) we investigate theeffect of size, 𝛽 , and the magnetic field on the transition energies ( 𝐸 − 𝐸 ) and ( 𝐸 − 𝐸 ) (Fig. 2). We observe that ( 𝐸 − 𝐸 ) decreases with increasing size, while ( 𝐸 − 𝐸 ) has almost no dependence on size. We fit ( 𝐸 − 𝐸 ) ∝ 1∕ 𝑅 𝛾 using Levenberg-Marquardt fit. The values of the exponent are found to be 𝛾 = 1 . for 𝛽 = 0 . and 𝛾 = 2 . for 𝛽 = 1 . This lateral size confinement effect be explained well by the asymptotic expression we developed in Sec. III.From Eq. (22), we note that the transition energies of interest are determined to be ( 𝐸 − 𝐸 ) ≈ 8 . ℏ 𝛽𝑚 𝑜 𝑅 ( √ 𝜎 ) − ℏ𝑒𝐵 𝑖 𝛽𝑚 𝑜 (24) ( 𝐸 − 𝐸 ) ≈ 2 ℏ𝑒𝐵 𝑖 𝛽𝑚 𝑜 (25)From Eq. (23) and Eq. (24), we note that ( 𝐸 − 𝐸 ) has a 𝑅 dependence on size, while ( 𝐸 − 𝐸 ) has no dependenceon size. We also examined the effect of 𝛽 on these transition energies. The transition energies have a strong dependenceon the magnitude of 𝛽 . As evident from Fig. 2, both transition energies increase sharply as 𝛽 is decreased. This isexpected from the 𝛽 dependence predicted by Eq. (23) and Eq. (24). Transition energy 𝐸 − 𝐸 increases two timeswhen magnetic field is increased ten times. However, the same transition energy has stronger dependence on BDDcondition. Energy ( 𝐸 − 𝐸 ) is increased seven times on decreasing 𝛽 ten times. Figure 3 shows the energy levels of our system as a function of the applied magnetic field. Interestingly the plotsare linear in 𝐵 . This can be understood on the basis of our asymptotic analysis (Eq. (22)). Cyclotron energy ( ℏ𝜔 𝑖 ∕2 )is manifestly linear in 𝐵 . Additionally a detailed analysis revealed that the first term in Eq. (22) ( √ 𝜎 ) is also linearin 𝐵 . Thus the exact results also indicate a linear trend of energy on magnetic field.We observe clear Landau level splitting as the magnetic field is increased. The effect of imposing BDD is againevident from the sharp fall in the energy levels for 𝛽 = 1 . We also find further support for our asymptotic approximation(represented by dashed lines), which converges well with our numerical results for large magnetic field strengths.Asymptotic equation (Eqs. (20) and (22)) points out an important fact. Changing the magnetic field outside does notradically affect the energy of the QD. This can be explained by the fact that probability of finding an electron in theouter region of the dot is very low. If 𝐵 𝑜 = 𝐵 𝑖 ∕ 𝛽 the effect of magnetic field in √ 𝜎 is zero . Which brings out the factthat we can use 𝐵 𝑜 as factor to make 𝐵 affect 𝐸 only in terms of the Landau energies. We carried a fit to energy of theform 𝐸 = 𝐶𝑅 𝛾 (26) First Author et al.:
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Page 5 of 9hort Title of the Article 𝐵 (T) 𝛽 𝛾 Table 1
Investigating the dependence of 𝐸 on 𝑅 as 𝐸 = 𝐶 ∕ 𝑅 𝛾 . Table gives the values of 𝛾 for various values of 𝛽 and 𝐵 . Table (1) lists the values of the exponent 𝛾 . Interestingly we find that BDD effect reduces 𝛾 . The magnetic field alsotends to reduce 𝛾 but marginally so. This can perhaps be understood as follows: increasing 𝐵 reduces the cyclotronradius of the electron and hence the electron density at the dot boundary is inconsequential. Drexler et. al. [21], in their experiment, used an In Ga As-GaAs QD ( 𝑚 𝑖 = 0 . 𝑚 𝑒 and 𝑚 𝑜 = 0 . 𝑚 𝑒 ,so 𝛽 ≈ 0 . ) with a radius of (10 ± 1) nm. They measured the transition gap ( 𝐸 − 𝐸 ) as a function of an appliedperpendicular magnetic field, and found their data to agree with a parabolic confinement model using 𝑚 ∗ = 0 . 𝑚 𝑜 and ℏ𝜔 𝑜 = 41 meV. Also, their dots are lens-shaped along the z-axis. However, they mention that their measured transitiongap data from IR spectroscopy is decoupled from the high energy z-confinement, and is only due to lateral confinement.The bandgap difference between In Ga As and GaAs is about meV. But this is further split between valenceand conduction band offsets, so a realistic value of the barrier height is closer to meV. In the present work, wehave developed a single-electron hard-wall confinement model including BDD within the effective mass framework.It is also important to note that our model is best applicable in a strong confinement regime described by Equations(8) and (9). Using 𝑉 = 100 meV and 𝛽 = 0 . , we find good agreement with their experimental data for a QD radiusof nm (Figure 4).In conclusion, we presented a hard-wall confinement based model for a circular Quantum Dot placed in a perpen-dicular magnetic field. Most importantly, we demonstrated the strong influence of using the BenDaniel-Duke boundarycondition on the set of spin-degenerate Landau levels of the system. We also developed a simple asymptotic approxi-mation (Eq. (22)) to the energy levels, which is in fair agreement with numerical results. The agreement is enhancedfor larger size and confinement potential. Finally, we also observed that a particular transition gap in our model agreeswell with experiments performed on the In Ga As-GaAs Quantum Dot [21].
5. ACKNOWLEDGEMENTS
We are extremely thankful to our mentor Praveen Pathak (HBCSE-TIFR) for regular discussions and guidance.We also also thank Vijay Singh (HBCSE-TIFR) for useful discussions. We acknowledge the support of the Govt. OfIndia, Department of Atomic Energy, under the National Initiative on Undergraduate Science (NIUS) of HBCSE-TIFR(Project No. 12-R&D-TFR-6.04-0600).
CRediT authorship contribution statement
Sriram Gopalakrishnan:
Formal analysis, Software, Visualization, Writing-Original Draft.
Sayak Biswas:
For-mal analysis, Software, Visualization.
Shivam Handa:
Formal analysis.
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Page 7 of 9hort Title of the Article [30] L. D. Landau, E. M. Lifshitz, Quantum mechanics: non-relativistic theory, Vol. 3, Elsevier, 2013.[31] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth edition Edition,Dover, New York, 1964.
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Page 8 of 9hort Title of the Article ( r ) = 0.1= 0.2= 0.5= 1.00 5 100246 (1, 0)(1, 1)(1, 2) Figure 1:
Scaled radial charge density ( 𝑟𝑔 ( 𝑟 ) ) profile of a QD with R = 10 nm, B = 1 T, and 𝑉 𝑜 = 1 eV. The main plotshows the effect of 𝛽 on the ground state charge density profile. The inset shows the charge density profile of levels (1,0),(1,1) and (1,2) for 𝛽 =1. ( 𝑚 𝑜 = 𝑚 𝑒 , 𝑚 𝑖 = 𝛽𝑚 𝑜 )First Author et al.: Preprint submitted to Elsevier
Page 9 of 9hort Title of the Article
10 12 14 16 18 20Size R (nm)1015202530354045 ( E E ) ( m e V ) = 0.7, numerical= 0.7, approx= 1.0, numerical= 1.0, approx10 15 20Size R (nm)3.54.04.55.0 ( E E ) ( m e V ) = 0.7= 1.0 Figure 2:
Transition energies as a function of the radius of the QD ( 𝑚 𝑜 = 0.067 𝑚 𝑒 , 𝑉 𝑜 = 100 meV, B = 1 T) in the rangeof 10 nm to 20 nm. The main plot shows the effect of size and 𝛽 on the transition gap ( 𝐸 − 𝐸 ) . The inset shows thesame for the transition gap ( 𝐸 − 𝐸 ) . The dashed lines in the main plot represent the asymptotic approximation (Eq.(22)). The case of 𝛽 = 0 . corresponds to the InGaAs-GaAs QD ( 𝑚 𝑖 = 0.047 𝑚 𝑒 , 𝑚 𝑜 = 0.067 𝑚 𝑒 ).First Author et al.: Preprint submitted to Elsevier
Page 10 of 9hort Title of the Article E n e r g y ( m e V ) (1,0), numerical(1,1), numerical(1,-1), numerical(1,2), numerical(1,-2), numerical(1,0), approx(1,1), approx(1,-1), approx(1,2), approx(1,-2), approx Figure 3:
Energy levels of the InGaAs-GaAs QD (R = 11 nm, 𝑉 𝑜 = 100 meV, 𝑚 𝑜 = 0.067 𝑚 𝑒 , 𝛽 = 0.7) as a function of anapplied magnetic field. The solid lines depict the lowest five levels as obtained from numerical computation. The dashedlines depict the same five levels as expected from our asymptotic approximation (Eq. (22))First Author et al.: Preprint submitted to Elsevier
Page 11 of 9hort Title of the Article ( E E ) ( m e V ) Our modelDrexler et. al.
Figure 4: